Chebyshev polynomials over finite fields and reversibility of -automata on square grids

Using number theory on function fields and algebraic number fields we prove results about Chebyshev polynomials over finite prime fields to investigate reversibility of two-dimensional additive cellular automata on finite square grids. For example, we show that there are infinitely many primitive irreversible additive cellular automata on square grids when the base field has order two or three.